AM$^3$: An Open-Source Tool for Time-Dependent Lepto-Hadronic Modeling of Astrophysical Sources

We present the AM$^3$ ("Astrophysical Multi-Messenger Modeling") software. AM$^3$ is a documented open source software that efficiently solves the coupled integro-differential equations describing the temporal evolution of the spectral densities of particles interacting in astrophysical environments, in-cluding photons, electrons, positrons, protons, neutrons, pions, muons, and neutrinos. The software has been extensively used to simulate the multi-wavelength and neutrino emission from active galactic nuclei (including blazars), gamma-ray bursts, and tidal disruption events. The simulations include all relevant non-thermal processes, namely synchrotron emission, inverse Compton scattering, photon-photon annihilation, proton-proton and proton-photon pion production, and photo-pair production. The software self-consistently calculates the full cascade of primary and secondary particles, including non-linear feedback processes and predictions in the time domain. It also allows to track separately the particle densities produced by means of each distinct interaction processes, including the different hadronic channels. With its efficient hybrid solver combining analytical and numerical techniques, AM$^3$ combines efficiency and accuracy at a user-adjustable level. We describe the technical details of the numerical framework and present three examples of applications to different astrophysical environments.